Summation of 1/p: Logarithmic Growth of N

[Dragonfall]

\sum_{p\leq N}\frac{1}{p}=\log\log N + A + O(\frac{1}{\log N})

Does it mean that we can simply replace the O part with a function that is a constant times 1/(log N)? What would be the difference between A + O(\frac{1}{\log N}) and O(1)?

 

[shmoe]

Does it mean that we can simply replace the O part with a function that is a constant times 1/(log N)?

No it doesn't. If f(n)=g(n)+O(h(n)) then there is a constant C where |f(n)-g(n)|<=C*h(n) in some suitable range of n. It does NOT mean f(n)=g(n)+C*h(n). Consider cos(x)=O(1) but we don't have cos(x)=constant.

What would be the difference between A + O(\frac{1}{\log N}) and O(1)?

The first gives more information (it implies the second but not vice versa). Even if you don't know the constant A (it can be expressed in terms of an infinite sum over the primes here though) it still says something about the structure of the lower order terms.